Analysis of multipacting threshold sensitivity to the random distributions of the secondary electron yield parameters

The way multipacting develops, depends strongly on the secondary emission property of the surface material. The knowledge of secondary electron yield is crucial for accurate prediction of the multipacting threshold. Variations in secondary electron yield parameters from experimental measurements create uncertainty, stemming from handling and surface preparation, and these uncertainties significantly affect multipacting threshold predictions. Despite their significance, the previous studies on the multipacting phenomenon did not adequately address the effect of an assumed random distribution of the secondary emission parameters on the multipacting threshold. Therefore, this paper aims to provide a comprehensive statistical study on how the different random distributions of the secondary emission parameters and, as a result, the uncertainty in the secondary electron yield affect multipacting thresholds. We focus on three commonly used distributions, namely uniform, normal, and truncated normal distributions, to define the uncertainty of random inputs. We use the chaos polynomial expansion method to determine how much each of the random parameters contributes to the multipacting threshold uncertainty. Additionally, we calculate Sobol sensitivity indices to evaluate the impact of the individual parameters or groups of parameters on the model outputs and study how different random distributions of these parameters affected the Sobol index results.


Appendix A
The steps of the gPC method are described in more detail:

Generation the polynomial expansion
The basis of the gPC approach is to provide a polynomial surrogate for the computational model.In this context, the polynomial expansion represents the relationship between the system's response (Y) and the independent input parameters ..... } in an M-dimensional space.The polynomial expansion can be expressed as following: (A-1(  is the multivariate polynomial, M is the total number of input parameters, N is the degree of the polynomial expansion, and α is the multi-index that indicate the degree of the polynomial in each of the input variables: Cα are the unknown coefficients to be determined.In our investigation, the system response we are interested in is the multipacting threshold, and we aim to approximate a function representation for this quantity.The total polynomial basis q during the surrogate modeling can be computed with the permutation formula: The multivariate polynomials are formed by the tensor products of the univariate orthogonal polynomials, which are as follows:  is the univariate orthogonal polynomial in the i th variable of degree αi 1 .

Determination of expansion coefficients
The coefficients Cj are determined by projecting the truncated expansion of Y on each basis polynomial and exploiting its orthogonality in the domain I, Where D()  is the probability density function (PDF) of the random parameters.α and β are the multiindices that indicate the degree of the polynomial in each of the input variables.There are many methods for numerically multidimensional integration or quadrature, which is a classical problem.Here is a generic illustration of numerical integration methods: where ξ (i), (ξi) are the integration points and the weights respectively, and n is the number of integration points.Y (ξi) is the value of Y in the integration point ξi.
In mathematics and numerical analysis, quadrature is an approximate method for computing integrals.
Error! Reference source not found.illustrates the comparison of integration techniques or quadrature and their respective integration points for three different random distributions.
To compute the coefficients CN in Eq. (A-5) and, consequently, approximate the quantity Y (X ( ),.....X ( )  , the deterministic model must be evaluated at the nodes (i) (i) of the sparse grid, where i range from 1 to n, representing the number of nodes.The numerical evaluation of the numerator in Eq. (A-5) can then be expressed as Eq.(A-7);  is the product of the corresponding weights.Once the ranges of deviations for the random parameter W1 are determined, the CST software is executed for each value within these ranges using python code.The SEY  is calculated for each random parameter, and the resulting SEY  are saved in a text file.These results will used to calculate the expansion coefficients in Equation (A-7).

Calculation of the statistical output
This approximate expansion of the function can be used to estimate statistical moments of the output, such as the mean and the variance, and to transfer the uncertainty of the input parameters to the output.

Accuracy assessment
Accuracy of the gPC is determined by comparing the approximation with the actual output of the model.The accuracy depends on the choice of orthogonal polynomials and the number of basic functions used in the expansion.In this study, we use a posteriori error estimation method to determine the relative error for the (N+1) st degree of the gPC, given by Eq .(A-8)as in 2 : (A-8) e

Global sensitivity analysis (Sobol's indices)
Sensitivity analysis assesses the influence of uncertain inputs parameters and interactions on the output variable (Y).A global, variance-based approach is valuable for customizing models by identifying inputs with minimal impact and quantifying the potential reduction in output uncertainty if these inputs were known.To achieve these objectives, Sobol introduce global, variance-based sensitivity indices.The first-order Sobol sensitivity index, also referred to as the main sensitivity index, quantifies the portion that Xi contributes directly (without interaction) to the total variance of the output V[Y].It aids to the identification of uncertain inputs that could be more precisely evaluated, thus facilitating input prioritization.The index is defined by Eq. (A-9); Where VXi represents the variance associated with the f (Xi), the term represents the conditional expectation, which signifies the anticipated value of the f(X) given a specific value of the uncertain input Xi.Parameter γα is given for each integration method in Table 3. Second-order sensitivity indices represent the portion of variance V[Y] resulting from Xi and Xj interaction, they are defined as Eq.(A-10); (A-10) VXi, Xj represents the variance associated with the Y (Xi, Xj).
In the following, we calculate the relative error of <SEY> for each degree of the gPC expansion for different uncertainty levels (5%,….30%).We presented the results for parameter W1 considering both normal and uniform distributions in Table A1.Based on Table A1, the expansion degree used for the gPC at which the relative error falls below 0.25%, increased with the level of uncertainty .For the uncertainty of 5%, 10% and 15%, the 5 th degree of expansion was used, and for the case of 20%, 25% and 30%, the 6 th and 7 th degrees of expansion were used.Furthermore, the obtained relative error of <SEY> using the Gussi-Hermit quadrature (normal distribution) is lower than that with the Clenshaw-Curtis quadrature (uniform distribution) used.

Appendix B
The statistical quantities related to the calculated values of <SEY> for different levels of uncertainties of r = 5, 10, 15, 20, 25, 30% of W1 are presented in

C
 is the expansion coefficient of the N th degree and N1 C   is the expansion coefficient of the (N +1) th degree.

Table A1 :
the gPC expansion degrees for different levels of uncertainty of W1

Table B1
TableB2shows that the multiplication threshold remains unchanged for random distributions for 15% and 20% uncertainty, but changes for 25% uncertainty with normal, uniform, and truncated normal